A classical approach to perform model discovery is by sparse regression and consists in finding \(\xi \)  such that, \(u_t = \Theta \cdot \xi \) , where \(u_t\)  is the time derivative of the field \(u\) . Each column of \(\Theta \)  is a candidate term of the underlying PDE, typically a combination of polynomial and spatial derivative functions (e.g. \(u\) , \(u_x\) , \(uu_x\) ). Usually, \(\xi \)  is identified based on a single experiment consisting of \(n\)  samples of the field \(u\) , see [1]}, [2]}, [3]}, [4]}, [5]}, [6]}. However, in practice observations of the same experiment might lead to some measurement differences. For instance, it might not be possible: (1) to fix identically the initial and/or boundary conditions of several experiments, (2) some of the parameters might not be controllable and will vary. As a result, there will be experimental results which exhibit natural variability. Let us denote \(q\)  the number of experiments at hand from which we would like to find the underlying partial differential equations. If we perform \(q\)  individual model discoveries as described earlier, the data across experiments will not be leveraged, loosing an opportunity for learning. To leverage the data across experiments, the \(q\)  libraries could be stacked \(\Theta = [\Theta _{1} , \Theta _{2}, \dots , \Theta _{q}]^T\)  and then a single sparse regression could reveal \(\xi \) , see [2]}, [8]}. However, this approach is not adapted if the coefficients of the underlying PDEs vary from an experiment to another.
In this paper, a more general approach to this is proposed: by promoting sparsity group-wise we leverage the data across experiments and are able to handle varying coefficients across datasets. In [9]}, a group sequentially thresholded ridge regression was proposed to infer parametric PDEs from a single dataset. In [10]} a group Iterative Hard Thresholding algorithm is used to enforce conservation laws and impose symmetries. In [11]} a group Lasso is used it to try to infer the law of gravitation from experimental ball drops.
